It depends on why you're releasing under GPL. If you're the sole rightsholder, and so not bound by the GPL, the obligations of GPLv3 s5a don't apply to you, and you can release without any particular text at all. If, however, you're required to release your work under GPLv3 by virtue of it being a derivative of a GPLv3 work, then s5a applies, which in turn invokes s4, which requires that "you conspicuously and appropriately publish on each copy an appropriate copyright notice".
We have previously argued that pseudonyms are acceptable, so this requirement is by no means onerous with regard to privacy, and I can't immediately think of any other reason why one should be reluctant.
The possible consequences of GPL violation are too complex and ramified to go into here, but you should avoid knowingly doing it if you can.
Should the main LICENSE file in a repository already contain the copyright, or should copyright be handled on a file-to-file basis?
There is nothing magical about a
LICENSE file, neither in copyright law, nor in most licences. Regarding copyright notices, the rubric of the GPL is clear that
It is safest to attach them to the start of each source file [...] and each file should have at least the “copyright” line and a pointer to where the full notice is found
Just because others (as you note) sometimes play a bit fast and loose with this, that is no reason not to follow best practice!